MinT - Best Known (105−30, 105, s)-Nets in Base 8

Best Known (105−30, 105, s)-Nets in Base 8

(105−30, 105, 484)-Net over F₈ — Constructive and digital

Digital (75, 105, 484)-net over F₈, using

1 times m-reduction [i] based on digital (75, 106, 484)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 30, 130)-net over F₈, using
    - trace code for nets [i] based on digital (0, 15, 65)-net over F₆₄, using
      - net from sequence [i] based on digital (0, 64)-sequence over F₆₄, using
        generalized Faure sequence [i]
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 0 and N(F) ≥ 65, using
        the rational function field F₆₄(x) [i]
        Niederreiter sequence [i]
  2. digital (45, 76, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 38, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(105−30, 105, 576)-Net in Base 8 — Constructive

(75, 105, 576)-net in base 8, using

t-expansion [i] based on (73, 105, 576)-net in base 8, using
- 7 times m-reduction [i] based on (73, 112, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 56, 288)-net in base 64, using
    - base change [i] based on digital (9, 48, 288)-net over F₁₂₈, using
      - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(105−30, 105, 3633)-Net over F₈ — Digital

Digital (75, 105, 3633)-net over F₈, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OA(8¹⁰⁵, 3633, F₈, 30) (dual of [3633, 3528, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8¹⁰⁵, 4096, F₈, 30) (dual of [4096, 3991, 31]-code), using
  - an extension C_e(29) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 8⁴âˆ’1, defining interval IÂ =Â [1,29], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 30 [i]

(105−30, 105, 1924404)-Net in Base 8 — Upper bound on s

There is no (75, 105, 1924405)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 66749 664669 073053 184648 224466 179945 094725 126939 985840 536831 477185 639970 457369 412072 986472 366672 > 8¹⁰⁵ [i]